Upgrade to hell
Getting an upgrade to business class on a flight to Amsterdam en route to NYC seemed splendid. But I soon discovered that I was upgraded to one of the two rows for business smokers.
The person on my left
The person to my left smoked Camel cigarettes. He had a pack of cigarettes on his table and he smoked a cigarette every 20 minutes. It was disturbing and I considered asking him not to smoke. But this was his right! He bought a business class ticket to the smoking section. I was just an upgrade, but even as a full-fledged businessman I would not have had the right to ask him not to smoke.
I had another idea. When breakfast was served I had a cup of coffee on my tray, and the pack of Camel cigarettes was on my neighbors’s tray bordering mine. If I were to accidentally spill all my coffee on his pack, this may take the cigarettes out of action!
It was not an easy decision. Will it work? Is it morally justified? Continue reading
This post complements Eran Nevo’s first post on the -conjecture
1) Euler’s theorem
Euler’s famous formula for the numbers of vertices, edges and faces of a polytope in space is the starting point of many mathematical stories. (Descartes came close to this formula a century earlier.) The formula for -dimensional polytopes is
The first complete proof (in high dimensions) was provided by Poincare using algebraic topology. Earlier geometric proofs were based on “shellability” of polytopes which was only proved a century later. But there are elementary geometric proofs that avoid shellability.
2) The Dehn-Sommerville relations
Consider a 3-dimensional simplicial polytope, – Continue reading
This post is authored by Eran Nevo. (It is the first in a series of five posts.)
What are the possible face numbers of triangulations of spheres?
There is only one zero-dimensional sphere and it consists of 2 points.
The one-dimensional triangulations of spheres are simple cycles, having vertices and edges, where .
The 2-spheres with vertices have edges and triangles, and can be any integer . This follows from Euler formula.
For higher-dimensional spheres the number of vertices doesn’t determine all the face numbers, and for spheres of dimension a characterization of the possible face numbers is only conjectured! This problem has interesting relations to other mathematical fields, as we shall see. Continue reading